Several results concerning the (optimal) regularity of the drift are available in order to have existence and uniqueness for rough differential equations. Namely, in [1]} one can see that whenever \(\sigma \in \mathcal {C}_b^{\lfloor p \rfloor +1}\)  and \(b\)  is globally Lipschitz continuous with linear growth, there is a unique solution to equation (REF ). Furthermore, some improvement of this criteria appears in [2]} (weak local Lipschitz condition and some control on the growth) and in [3]} (Lyapounov conditions on \(b\) ). In order to go beyond (using some stochasticity of the rough path), one needs to develop an other approach of the solution to the RDE with drift. Following a classical ideas of flow transform (see for example [2]}) we give an equivalent formulation for the solutions of Equation (REF ).
